187edt
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| ← 186edt | 187edt | 188edt → |
187 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 187edt or 187ed3), is a nonoctave tuning system that divides the interval of 3/1 into 187 equal parts of about 10.2 ¢ each. Each step represents a frequency ratio of 31/187, or the 187th root of 3.
Theory
187edt is nearly identical to 118edo, but with the perfect twelfth rather than the octave being just. The octave is stretched by about 0.164 cents. Like 118edo, 187edt is consistent to the 12-integer-limit. It preserves the 5-limit microtempering quality of 118edo, and the approximated prime harmonics 7, 11, 17, and 19 are slighly improved.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.16 | +0.00 | +0.33 | +0.51 | +0.16 | -2.26 | +0.49 | +0.00 | +0.67 | -1.60 | +0.33 |
| Relative (%) | +1.6 | +0.0 | +3.2 | +5.0 | +1.6 | -22.3 | +4.8 | +0.0 | +6.6 | -15.7 | +3.2 | |
| Steps (reduced) |
118 (118) |
187 (0) |
236 (49) |
274 (87) |
305 (118) |
331 (144) |
354 (167) |
374 (0) |
392 (18) |
408 (34) |
423 (49) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +4.15 | -2.10 | +0.51 | +0.66 | -2.59 | +0.16 | -1.90 | +0.84 | -2.26 | -1.43 | +2.98 | +0.49 |
| Relative (%) | +40.8 | -20.6 | +5.0 | +6.5 | -25.5 | +1.6 | -18.7 | +8.2 | -22.3 | -14.1 | +29.3 | +4.8 | |
| Steps (reduced) |
437 (63) |
449 (75) |
461 (87) |
472 (98) |
482 (108) |
492 (118) |
501 (127) |
510 (136) |
518 (144) |
526 (152) |
534 (160) |
541 (167) | |
Subsets and supersets
Since 187 factors into primes as 11 × 17, 187edt contains 11edt and 17edt as subset edts.